 Solution

Use the module criterion: an element $x$ is an algebraic integer if and only if it lies in a nonzero finitely generated $\mathbb{Z}$-module $M$ with $xM\subseteq M$.

Let the coefficients of the monic polynomial be $c_0,\dots ,c_{n-1}$ and put $A=\mathbb{Z}[c_0,\dots ,c_{n-1}]$. A ring generated by finitely many algebraic integers is a finitely generated $\mathbb{Z}$-module. If $\alpha$ is a root, thenis finitely generated, and the monic relationshows that $\alpha M\subseteq M$. The criterion proves that every root is an algebraic integer.